Existence and Uniqueness of Solution for Perturbed Nonautonomous Systems with Nonuniform Exponential Dichotomy

Yong-Hui Xia, Xiaoqing Yuan, Kit Ian Kou and Patricia J. Y. Wong

Abstract and Applied Analysis, 2014, vol. 2014, issue 1

Abstract: Nonuniform exponential dichotomy has been investigated extensively. The essential condition of these previous results is based on the assumption that the nonlinear term satisfies |f(t, x)| ≤ μe−ε|t|. However, this condition is very restricted. There are few functions satisfying |f(t, x)| ≤ μe−ε|t|. In some sense, this assumption is not reasonable enough. More suitable assumption should be |f(t, x)| ≤ μ. To the best of the authors′ knowledge, there is no paper considering the existence and uniqueness of solution to the perturbed nonautonomous system with a relatively conservative assumption |f(t, x)| ≤ μ. In this paper, we prove that if the nonlinear term is bounded, the perturbed nonautonomous system with nonuniform exponential dichotomy has a unique solution. The technique employed to prove Theorem 4 is the highlight of this paper.

Date: 2014
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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnlaaa:v:2014:y:2014:i:1:n:725098

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